Topic 2 - Logarithmic And Exponential

1 TOPIC 2: EXPONENTIAL AND LOGARITHMIC FUNCTIONS 2.1 : Relationship Between Exponential And Logarithmic Functions Example 1 Convert the following to logarithm form: 1 a) 23 = 8 (b) 3-2 = 9

(c) 2x = 47

Example 2 Convert the following to exponential form/ index form: a) log2 32 =5 (b) log3 27 = 3 (c) log2 y = x Example 3 Find the value of each of the following: (a) log 2 64 (b) log 3 1 (e) log3  1  (f) log 8 0.25  81  The important property is

(c) log 7 7

y  ax

a  0 , a 1

x  log a y ,

y  0, a  0, a  1

(d) log4 16-1

log a 1  0, log a a  1, log a a b  b, log a  a  undefined / no solution 2.2: Properties of Logarithms Law of logarithms. There are four basic laws of logarithms. (1) log a mn  log a m  log a n m  log a m  log a n n (3) log a m n  n log a m

(2) log a

(4)log a b 

For two logarithms of the same base, loga M = loga N Then, M = N

log c b log c a

Example 4 Simplify the following, expressing each as a single logarithm: (a) log 2 4 + log 2 5 – log210 (b) 2log a 5 – 3 log a 2 (c) log 8 4 + log 2 16 Example 5 If log 2 = r and log 3 = s, express in terms of r and s (a) log 16 (b) log 18 (c) log 13.5

2 Exercise 2.1: Logarithmic and Exponential functions 1. Write each of the following in the form y = bx. (a) log28 = 3 (b) log381 = 4 (c) log50.04 = -2 (d) log7x = 4 (e) logx5 = t (f) logpq = r 2. (a) (b) (c) (d) (e) (f)

Write each of the following in the form x = logby. 23 = 8 36 = 729 4-3 = 641 a8 = 20 h9 = g mn = p

3. Evaluate the following: (a) log216 (b) log416 (c) log7 1 49 (d) log41 (e) log55 (f) log27 1 3 (g) log168 (h) log3 2 2 (i) log 2 8 2 4. (a) (b) (c) (d) (e) (f) (g)

Find the value of y in each of the following. logy 49 = 2 log4 y = -3 log3 81 = y log10 y = -1 log2 y = 2.5 logy 1296 = 4 log 1 y  8 2

(h) log 1 1024  y 2

(i) logy 27 = -6 Page 34, Exercise 3A: Question 6, 7, 8, 9 (a) – (e)

3 Exercise 2.2: Law of logarithms 1. Write each of the following in terms of log p, log q and log r. The logarithms have base 10. (a) log pqr (b) log pq2r3 (c) log 100pr5 p (d) log q 2r pq (e) log 2 r 1 (f) log pqr p (g) log r qr 7 p (h) log 10 10 p 10 r (i) log q

2. Express as a single logarithm, simplifying where possible. (All the logarithms have base 10, so, for example, an answer of log 100 simplifies to 2.) (a) 2log 5 + log 4 (b) 2log 2 + log 150 – log 6000 (c) 3log 5 + 5 log3 (d) 2log 4 – 4log 2 (e) log 24  12 log 9  log 125 (f) 3log 2 + 3log 5 – log 106 (g) 12 log 16  13 log 8 (h) log 64 – 2log 4 + 5log 2 – log 27

3. If log 3 = p, log 5 = q, log 10 = r, express the following in terms of p, q and r. (All the logarithms have the same unspecified base.) (a) log 2 (b) log 45 (c) log 90 (d) log 0.2 (e) log 750 (f) log 60 (g) log 16 (h) log 4.05 (i) log0.15 Page 36, Exercise 3B: 1, 2, 3

4

x

2.3 The relationship between a and logax Exponential function is y = ax. Logarithmic function is y = logax. The graphical relationship between y = ax and y = logax is the reflection in the line y = x.

The gradient of y = ex is ex. The gradient of y = ln x is x -1. The functions ex and ln x are inverse functions, the graph of y = ex and y = ln x are mirror images in the line y = x. The range of f : x  e x for x  R is f( x)  R  . The range of f : x  ln x for x  R  is f( x)  R .

2.4

Exponential Equations and inequalities

Properties of Indices: (1) (2) (3) (4)

2a × 2b = 2a + b 2a ÷ 2b = 2a – b 2a × 3a = (2×3)a = 6a (2a)b = 2a× b

Example 6: Without using table or calculator, solve the equations: (b) 2 × 4x + 1 = 1612x (c) 4x – 9(2x) + 8 = 0 (d) ex – e-x = 0 (a) 4x ×32x = 216 Example 7: Given y = axb and y = 2 when x = 3, y  Example 8: Solve the following inequalities: (a) 2x < 16 (b) 4n > 750

2 when x = 9, find a and b. 9

(c) (0.6)n < 0.2

Example 9: Find the smallest value of n for which the nth term of the geometric progression with first term 2 and common ratio 0.9 is less than 0.1. Example 10: How many terms of the geometric series 1 + 2 + 4 + 8 + … must be taken for the sum to exceed 1011?

5 Exercise 2.3: Solving exponential equations and inequalities 1. Without using table or calculator, solve the following equations: (a) 2x × 5x = 1000 (h) 4 x  32 x  6 2x x (b) 3 × 4 = 36 (i) .53 x  25 x 1  1 2 125 3 y 3  9 2 y (c) (d) (e) (f) (g) 2.

2x + 2 – 3 = 5 × 2x - 1 32x + 1 – 28(3x) + 9 = 0 42x – 68 (4x) + 256 = 0 32x + 1 – 82(3x) + 27 = 0

(j) (k) (l)

.. 3 x  3 x  2  90 5  2x  2  4x  2 2 2 x 3  2 x 3  1  2 x

The curve y = abx passes through (1, 96), (2, 1152) and (3, p). Find the values of a, b and p.

3. The curve y = axn passes through (2, 9) and (3, 4). Calculate the values of a and n. 4. Given that y = axb – 5, and that y = 7 when x = 2, and y = 22 when x = 3, find the values of a and b. 5. Solve the simultaneous equations: 3 x  9 2 y  27 1 8 6. Solve the following inequalities: (a) 5x ≥ 125 (b) 0.4n > 0.45 (c) 2x > 128 (d) 3x ≤ 243 (e) 7x ≤ 49-1 2x  4y 

7. Find the smallest possible integer n such that (a) 2n > 104 (b) 2n > 50 (c) 5.2n > 1000 (d) 0.5n < 10- 4 8. Find the largest possible integer n such that (a) 2.7n < 850 (b) 6.2n < 4000 (c) 4.6n < 30000 9. Find the least number of terms in the Geometric progression, 2 + 2.4 + 2.88 + … such that the sum exceeds 1 million. 10. How many terms of geometric series 2 + 6 + 18 + 54 + … must be taken for the sum to exceed 3 million? 11. A biological culture contains 500 000 bacteria at 12 noon on Monday. The culture increases by 10% every hour. At what time will the culture exceed 4 million bacteria?

6 2.5: Logarithmic equations Properties of logarithms: (1) log a mn  log a m  log a n m  log a m  log a n n (3) log a m n  n log a m

(2) log a

(4)log a b 

log c b log c a

Example 11 Solve the equation log2(x – 1) + log2(x + 3) - log2(x + 1) = 1 Example 12 Solve the equation log2

x + 2 log2 x – 2 = 0

Example 13 Solve the simultaneous equation:

log2(x – 4y) = 4 log84x – log8(8y + 5) = 1

Example 14 Solve the following logarithmic equations: (b) (a) log 3 N + log 9 N = 6

log 5 x = 4 log x5

Exercise 2.4: Logarithmic equations 1. Solve the following equations without using calculator. (a) lg 4 + 2 lg x = 2 (b) log2y2 = 3 + log2(y + 6) (c) lg y + lg (2y – 1) = 1 (d) loga7 + logax = 0 2. Without using calculator, solve (a) lg25 + lg x – lg(x – 1) = 2 (b) 2lg 3 + lg2x – lg(3x +1) = 0 (c) logy8 = ½ (d) 2log2y = 4 + log2(y + 5) (e) lg(x2 + 12x – 3) = 1 + 2lgx 3. Solve the simultaneous equations: (a) 3x = 9(27)y log27 – log2(11y – 2x) = 1 (b) lg x + 2lgy = 3 x2y = 125

7 2.6: Linear law: Using logarithm to transform curves into linear lines Convert (a) the equation y = axn to logarithmic form, giving a straight line when log y is plotted against log x. (b) the equation y = A(bx) to logarithmic form, giving a straight line when log y is plotted against x. (c) the equation y = Aenx to the form ln y = ln A + nx, giving a straight line when ln y is plotted against x.

Example 15 Jack takes out a fixed rate savings bond. This means he makes one payment and leaves his money for a fixed number of years. The value of his bond, $B, is given by the formula B = Axn where A is the original investment and n is the number of complete years since he opened the account. The table gives some values of B and n. By plotting a suitable graph find the initial value of Jack’s investment and the rate of interest he is receiving. n 2 3 5 8 10 B 982 1056 1220 1516 1752

Example 16 The figure shows part of a straight line graph obtained by plotting values of the variables indicated. Express y in terms of x.

8 Exercise 2.5: Using logarithm to transform curves into linear lines 1. From the given graph (i), find (a) lg y in terms of lg x

(b) y in terms of x

2. Given the relationship for (ii) is y = axb, find (a) the values of a and b (b) y when x = 5

3. (a) If log10 y = 0.4 + 0.6x, express y in terms of x. (b) if log10y = 0.7 + 2 log10 x, express y in terms of x. 4. The table shows the mean relative distance, X, of some of the planets from the Earth and the time, T years, taken for one revolution round the sun. By drawing an appropriate graph show that there is an approximate law of the form T = aX n, stating the values of a and n. Mercury Venus Earth Mars Saturn X 0.39 0.72 1.00 1.52 9.54 T 0.24 0.62 1.00 1.88 29.5

5. In a spectacular experiment on cell growth the following data were obtained, where N is the number of cells at a time t minutes after the start of the growth. t N

1.5 9

2.7 19

3.4 32

8.1 820

10 3100

At t = 10 a chemical was introduced which killed off the culture. The relationship between N and t was thought to be modeled by N = abt, where a and b are constants. (a) Use a graph to determine how these figures confirm the supposition that the relationship is of this form. Find the values of a and b, each to the nearest integer. (b) If the growth had not been stopped at t = 10 and had continued according to your model, how many cells would there have been after 20 minutes? (c) An alternative expression for the relationship is N = m ekt. Find the values of m and k. Page 46, Ex 3D: 1 (a), 5, 7